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Appendix VL                           1079

AI B! by means of opposing couples that rotate relatively through the
angle 0 = ?2 .    Each couple is—

E[                            El

Bm = —         or a force of —5- at an arm p

/>                                                p*                            r

This force is exerted, during rotation, through a space of 20 ft.

1     A              EI

Apparent work done = —5- x 20 =

rt                                         |0-                  |02I2

30,000,000 x 60 x i x 20           0        .

~          72X72x12      ""  =   55S,ooo foot pounds

on the assumption that the stress and strain are within the elastic
limit of the material. Examining further,

Total circumference at neutral axis = 2 x ?r x 72 = 452*55
Total outside circumference = 2 x?rX72i = 455'55
Extension on circumference = 3 ins.

Extension on 20 ft. length of plate = — -- -- = rco ins.

452            ^J

Extension per cent. = *66 (apparent).

Also stress in tons per sq. in.

-     Ey       30,000,000 x *5                f             .

= /-T =    6x12x2240     - 93-5 (apparent).

Taking co - ordinates, make EF = "66 % and DF = 93*5 tons.
Join ED, the elastic line. Refer now to diagram E, Fig. 345, p. 386,.
and plot it within the triangle DEF, Fig. 956, to form the shaded area.
We now see that the apparent work estimate must be considerably
modified, for the elastic tension cannot exceed "15%, and the real
work will not be represented by the area DEF as was supposed, but
by the shaded area. Now

Area DEF a 93.? 66 a 300o

Shaded area a ^       X23 a 1320

01       i  j           1320x588,000                 r   ^         ,

.*. Real work done =   -~ — -g-— «- -   = 254,000 foot pounds,

jP.  429.     Moment   of   Resistance   by  Calculation.

Referring to Fig. 957, let ol be the axis, at right angles to the
paper, of the centroid of an area, and let a small element a of the
area be placed at ?. Let the 2nd moment round o be called
\ and that round O2 an axis parallel to that through ol be called